Problem: Circle $T$ has a circumference of $12\pi$ inches, and segment $XY$ is a diameter. If the measure of angle $TXZ$ is $60^{\circ}$, what is the length, in inches, of segment $XZ$?

[asy]
size(150);
draw(Circle((0,0),13),linewidth(1));
draw((-12,-5)--(-5,-12)--(12,5)--cycle,linewidth(1));
dot((0,0));

label("T",(0,0),N);
label("X",(-12,-5),W);
label("Z",(-5,-12),S);
label("Y",(12,5),E);

[/asy]
We can begin by using the circumference to solve for the radius of the circle.  If the circumference is $12\pi$, then $2\pi r=12\pi$ which implies $r=6$.  Now, we can draw in the radius $TZ$ as shown: [asy]
size(150);
draw(Circle((0,0),13),linewidth(1));
draw((-12,-5)--(-5,-12)--(12,5)--cycle,linewidth(1));
draw((0,0)--(-5,-12),linewidth(1)+linetype("0 4"));
dot((0,0));

label("T",(0,0),N);
label("X",(-12,-5),W);
label("Z",(-5,-12),S);
label("Y",(12,5),E);

[/asy]

We know that $TX=TZ$, since both are radii of length 6.  We are given $\angle TXZ=60^{\circ}$, so $\angle TZX=60^{\circ}$, and triangle $TXZ$ is equilateral.  Thus, $TX=TZ=XZ=\boxed{6}$.